Knowing that a span is the space spanned by a set of vectors, we might want to go the other way and find a set of vectors that describe the space. That set of vectors will be called a basis.
Finding that set only requires that we impose two conditions we already know about on a set of vectors.
- The set of vectors must span the space we want to describe.
- The set of vectors must be linearly independent: that is to say that there must not be any redundant vectors in the set.
With these definitions in mind, we can already make some observations about the basis.
- An -dimensional space must have at least vectors in its basis, such that it could span the entire space.
- An -dimensional space must have at most vectors in its basis such that no vector is linearly dependant on another.
Generally speaking, if you have some -dimensional space then it should always be possible to, using vectors that form a basis, reach every possible point in that space with linear combinations of those vectors.
It should come as no surprise that in 3-dimensional space, the unit vectors are a basis for the three dimensional space. It should be obvious that by scaling each of them by a different constant and adding them all together that any point in that space can be reached.
Because of that, we have special names for them: , and .
The unit vectors are not the only valid basis for an n-dimensional space, however. Those vectors could be scaled by any amount and they would still be a basis, where we could combine those three to reach any other vector.
In fact, you could imagine squeezing all of those vectors in towards a line and as long as they all point in different directions, it is still possible to combine them in such a way such that it is possible to reach any point in the space with them.
It makes more sense if you think about what happens to the rest of space if you apply the same change in the basis vectors to every other part of the space.
It is only when all of space is flattened on to a line or a plane that those vectors stop being a basis for the rest of the space.